GM Lesson 062 Practical Problems Involving Linear Equations

Learning Intentions

By the end of this lesson, students will be able to:

• Use linear equations to represent practical problems.
• Solve equations to find unknown quantities.
• Interpret solutions using appropriate units and context.

Prerequisites

Students should already be able to:

• Solve one-step and two-step linear equations.
• Solve equations with variables on both sides.
• Translate descriptions in words into linear equations.
• Check a solution by substituting it into the original equation.

Key Idea Summary

A practical problem can often be solved by defining an unknown quantity, writing a linear equation, solving it, and interpreting the answer in context.

A useful structure is:

1. Define the unknown.
2. Write an equation.
3. Solve the equation.
4. Check that the answer is reasonable.
5. State the answer using correct units.

A linear equation is useful when a situation involves a fixed amount plus or minus a repeated amount.

For example:

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

When solving practical problems involving linear equations, students should first identify what is unknown.

Common unknowns include:

• the number of items
• the number of hours
• the number of kilometres
• the original amount
• the cost per item
• the number of people

The unknown should be represented using a pronumeral.

For example:

Let be the number of hours worked.

Then write an equation that represents the information in the problem.

Worked Example 1: Taxi Fare

A taxi charges a fixed booking fee of $ plus $ per kilometre. A passenger pays $ in total. How many kilometres did the taxi travel?

Let be the number of kilometres travelled.

The total cost is:

Since the passenger pays $ :

The taxi travelled kilometres.

Check:

The answer is reasonable because the fixed fee plus kilometres gives the total fare.

Worked Example 2: Gym Membership

A gym charges a joining fee of $ and then $ per week. After several weeks, the total cost is $ . How many weeks has the person been a member?

Let be the number of weeks.

The total cost is:

So:

The person has been a member for weeks.

Check:

So the solution is correct.

Worked Example 3: Perimeter Problem

A rectangle has a length of cm and a width of cm. Its perimeter is cm. Find the width and length.

The perimeter of a rectangle is:

Here:

and

So:

The width is cm.

The length is:

So the rectangle has width cm and length cm.

Check:

Worked Example 4: Comparing Two Payment Options

A streaming service offers two plans.

Plan A costs $ per month plus a setup fee of $ .

Plan B costs $ per month with no setup fee.

After how many months will the two plans cost the same?

Let be the number of months.

Plan A costs:

Plan B costs:

Set the costs equal:

The two plans cost the same after months.

Check:

Plan A:

Plan B:

Both plans cost $ after months.

Understanding Checks

Check 1

A parking garage charges $ plus $ per hour. A driver pays $ . Write an equation to find the number of hours parked.

Expected equation:

Check 2

Solve the equation from Check 1.

The driver parked for hours.

Check 3

A number is multiplied by and then is added. The result is . Write and solve an equation.

Let be the number.

The number is .

Check 4

A phone company charges $ per month plus $ per text message. A monthly bill is $ . What does the solution represent?

Equation:

Solution:

The solution means text messages were sent.

Exercises

Simple Familiar Exercises

Exercise 1

A delivery company charges a fixed fee of $ plus $ per kilometre. The total cost is $ .

a. Define the unknown.
b. Write a linear equation.
c. Solve the equation.
d. Interpret the answer.

Exercise 2

A cinema ticket booking includes a service fee of $ plus $ per ticket. The total cost is $ .

a. Let be the number of tickets. Write an equation.
b. Solve for .
c. State the answer in context.

Exercise 3

A person earns $ per hour. One week they earn $ .

a. Write an equation for the number of hours worked.
b. Solve the equation.
c. Interpret the solution.

Exercise 4

A rectangle has width cm and length cm. Its perimeter is cm.

a. Write an equation using the perimeter.
b. Solve for .
c. Find the width and length.

Exercise 5

A number is doubled and then is subtracted. The result is .

a. Define the unknown.
b. Write an equation.
c. Solve the equation.

Exercise 6

A gym charges $ per week plus a joining fee. After weeks, the total cost is $ .

a. Let be the joining fee. Write an equation.
b. Solve for .
c. Interpret the solution.

Complex Familiar Exercises

Exercise 7

A plumber charges a call-out fee of $ and then $ per hour. A customer pays $ .

a. Write an equation for the number of hours worked.
b. Solve the equation.
c. Explain why the answer is reasonable.

Exercise 8

Two car hire companies offer different pricing.

Company A charges $ plus $ per kilometre.

Company B charges $ plus $ per kilometre.

a. Let be the number of kilometres driven. Write an expression for each company.
b. Write an equation to find when the costs are equal.
c. Solve the equation.
d. Interpret the result.

Exercise 9

A school sells adult tickets and student tickets to a concert. Adult tickets cost $ and student tickets cost $ . A family buys adult ticket and several student tickets for a total of $ .

a. Let be the number of student tickets. Write an equation.
b. Solve the equation.
c. State the number of student tickets bought.

Exercise 10

A rectangle has length metres and width metres. The perimeter is metres.

a. Write an equation using the perimeter.
b. Solve for .
c. Find the length and width.
d. Check the perimeter.

Exercise 11

A mobile phone plan charges a fixed monthly fee of $ plus $ per extra gigabyte of data. A customer is charged $ for the month.

a. Write an equation for the number of extra gigabytes used.
b. Solve the equation.
c. Interpret the result.

Exercise 12

A fundraiser sells raffle tickets. The group spends $ on prizes and printing. Each ticket is sold for $ . They want to make a profit of $ .

a. Write an equation for the number of tickets that must be sold.
b. Solve the equation.
c. Explain what the solution means.

Homework Problems

Problem 1

A taxi charges $ plus $ per kilometre. The total fare is $ .

a. Define the unknown.
b. Write an equation.
c. Solve the equation.
d. Interpret the answer.

Problem 2

A streaming platform charges $ per month after an initial setup fee. After months, the total cost is $ .

a. Let be the setup fee. Write an equation.
b. Solve for .
c. State the answer in context.

Problem 3

A rectangle has width cm and length cm. Its perimeter is cm.

a. Write an equation.
b. Solve for .
c. Find the width and length.

Problem 4

A personal trainer offers two payment options.

Option A: $ per session plus a $ registration fee.

Option B: $ per session with no registration fee.

a. Let be the number of sessions. Write an expression for each option.
b. Write an equation to find when the two options cost the same.
c. Solve the equation.
d. Interpret the result.

Problem 5

A school camp costs $ plus $ per activity. A student pays $ in total.

a. Write an equation for the number of activities.
b. Solve the equation.
c. Check your answer by substitution.

Problem 6

A number is divided by and then is added. The result is .

a. Define the unknown.
b. Write an equation.
c. Solve the equation.
d. Check the solution.

Next: GM Lesson 063 Choosing and Checking Linear Equation Models